Complex roots

Given a complex w and a positive natural number n, we say that z is the nth root of w if zⁿ=w. We set:

ⁿ√w = { z ∈ ℂ : zⁿ=w}

that is, ⁿ√w is the set of n-th roots of w. From the fundamental theorem of algebra there are exactly n n-th roots of w so | ⁿ√w |=n. Note that with this definition it makes sense to calculate for example √-1 and more generally the “root of a negative number” which is absolutely not allowed in IR. Let us calculate √-1 as an example. Since -1=e^(iπ) it follows that if z=ρe^(iθ) is a complex such that z²=-1 then ρ²e^(i2θ)=e^(iπ) so ρ=1 and 2θ=π+2kπ with k=0,1,2,…from which we deduce (for successive values of k we obtain the same complexes):

√-1 = {e^(iπ/2),e^(3π/2)}={i,-i}

in less precise terms we say that √-1 = i or √-1=-i.